1. There are several equivalent definitions of concavity. (1) We say a twice-differentiable function f to be (strictly) concave up on (a, b) if for any distinct T1, T2 (a, b)(assume x1 < x2) and for any t = (0, 1) we have f(x1+t(x2 − x1)) < f(x1)+t(f(x2) − f(x1)). (2) We say a twice-differentiable function f to be (strictly) concave up on (a, b) if for any distinct T1, T2 € (a, b)(assume x1 < x2) and for any 1 € (11, 12), f(x) f(x1) f(x2) f(x1) f(x2)- f(x) x-x1 < I2X1 < X2-X (3) We say a twice-differentiable function f to be (strictly) concave up on (a,b) if f' is (strictly) increasing on (a,b). (a) Show that definition (1) is equivalent to definition (2). Hint: You may want to use this fact: if b> 0, d > 0 and < a a+c C < b b+d d <, we have

1. There are several equivalent definitions of concavity. (1) We say a twice-differentiable function f to be (strictly) concave up on (a, b) if for any distinct T1, T2 (a, b)(assume x1 < x2) and for any t = (0, 1) we have f(x1+t(x2 − x1)) < f(x1)+t(f(x2) − f(x1)). (2) We say a twice-differentiable function f to be (strictly) concave up on (a, b) if for any distinct T1, T2 € (a, b)(assume x1 < x2) and for any 1 € (11, 12), f(x) f(x1) f(x2) f(x1) f(x2)- f(x) x-x1 < I2X1 < X2-X (3) We say a twice-differentiable function f to be (strictly) concave up on (a,b) if f' is (strictly) increasing on (a,b). (a) Show that definition (1) is equivalent to definition (2). Hint: You may want to use this fact: if b> 0, d > 0 and < a a+c C < b b+d d <, we have

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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